söndag 10 november 2013

More on Standard Calculus as Backward Magic

The Backward Magic aspect of Standard Calculus is expressed by the fact that a main role of the Fundamental Theorem is to compute the area $A(a,b)$ under the graph of a real-valued function $v:[a,b]\rightarrow R$ defined on an interval $[a,b]$, by finding a primitive function $x(t)$ of $v(t)$ satisfying $\frac{dx}{dt} = v$, and then computing

$A(a,b) = x(b) - x(a)$.

Here $v(t)$ can be momentary velocity and $x(t)$ traveled distance from some position. The laborious work of computing the area $A(a,b)$ by summing the contributions over a partition of $[a,b]$ into many small intervals, that is computing a Riemann sum, is thus avoided and magically replaced by simply evaluating the difference $x(b) - x(a)$. This was the magic which could be performed by Leibniz and Newton in front of a stunned audience at the end of the 17th century, and this is the trick each standard Calculus teacher performs today in front of a mystified class of students.

But the magic was based on somehow analytically finding a primitive function, that is by solving the differential equation $\frac{dx}{dt} =v$ analytically. This could and can be done for certain functions $v$, but the analytical machinery may be very involved and often simply impossible. The natural generalization to $v$ depending on $x$ is even more difficult analytically.

Today the computer can solve the equation $\frac{dx}{dt}=v$ by time-stepping corresponding to computing a Riemann sum as an approximation of the area, and there is no need to resort to the magics of finding a primitive function analytically. The generalization to $v$ depending on $x$ is direct and easy by time-stepping with computer. What was difficult to Leibniz and Newton, and largely motivated Calculus, is easy for the student today. This gives Calculus a different meaning as Forward Rational time-stepping, which is not the Backward Magic offered students of standard Calculus.